New paper

E. I. Khukhro and P. Shumyatsky, Almost Engel finite and profinite groups, submitted, 2015; arXiv:1512.06097.

Abstract: Let $g$ be an element of a group $G$ . For a positive integer $n$ , let $E_n(g)$ be the subgroup generated by all commutators $[...[[x,g],g],\dots, g]$ over $x\in G$ , where $g$ is repeated $n$ times. We prove that if $G$ is a profinite group such that for every $g\in G$ there is $n=n(g)$ such that $E_n(g)$ is finite, then $G$ has a finite normal subgroup $N$ such that $G/N$ is locally nilpotent. The proof uses the Wilson–Zelmanov theorem saying that Engel profinite groups are locally nilpotent. In the case of a finite group $G$ , we prove that if, for some $n$ , $|E_n(g)|\leq m$ for all $g\in G$ , then the order of the nilpotent residual $\gamma _{\infty} (G)$ is bounded in terms of $m$ .